1 / 21

BCOR 1020 Business Statistics

BCOR 1020 Business Statistics. Lecture 17 – March 18, 2008. Overview. Chapter 8 – Sampling Distributions and Estimation Confidence Intervals Mean ( m ) with variance ( s ) unknown Sample size determination for the mean ( m ). Chapter 8 – Confidence Interval for a Mean ( m ) with Unknown s.

bunme
Download Presentation

BCOR 1020 Business Statistics

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. BCOR 1020Business Statistics Lecture 17 – March 18, 2008

  2. Overview • Chapter 8 – Sampling Distributions and Estimation • Confidence Intervals • Mean(m) with variance (s) unknown • Sample size determination for the mean (m)

  3. Chapter 8 – Confidence Interval for a Mean (m) with Unknown s Suppose we want to find a confidence interval for m when s is unknown: • Instead of using • We will estimate s with S and use We need to know how this statistic distributed to compute the confidence interval for m when s is unknown?

  4. Chapter 8 – Confidence Interval for a Mean (m) with Unknown s has a Student’s t distribution with (n – 1) degrees of freedom, denoted T ~ t(n–1). • Assumes the population from which we are sampling is Normal. • t distributions are symmetric and shaped like the standard normal distribution. • The t distribution is dependent on the size of the sample. (As n increases, the t distribution approaches the standard normal.)

  5. Chapter 8 – Confidence Interval for a Mean (m) with Unknown s Degrees of Freedom: • Degrees of Freedom (d.f.) is a parameter based on the sample size that is used to determine the value of the t statistic. • Degrees of freedom (denoted by n) tell how many independent observations are used to calculate our estimate of s, S. n = n - 1 • For a given confidence level, t is always larger than z, so a confidence interval based on t is always wider than if z were used.

  6. Chapter 8 – Confidence Interval for a Mean (m) with Unknown s Comparison of z and t: • For very small samples, t-values differ substantially from the normal. • As degrees of freedom increase, the t-values approach the normal z-values. • For example, for n = 31, the degrees of freedom are: • What would the t-value be for a 90% confidence interval? n = 31 – 1 = 30

  7. Chapter 8 – Confidence Interval for a Mean (m) with Unknown s Comparison of z and t: • For n = 30, the corresponding t-value is 1.697. • Compare this to the 90% z-value, 1.645.

  8. Clickers Use the table below to find the t-value for a 95% confidence interval from a sample of size n = 6. A = 3.365 B = 2.015 C = 1.960 D = 2.571

  9. sn x+ta Chapter 8 – Confidence Interval for a Mean (m) with Unknown s • Use the Student’s t distribution instead of the normal distribution when the population is normal but the standard deviation s is unknown and the sample size is small. Constructing the Confidence Interval with the Student’s t Distribution: • Solving the inequality inside the probability statement • P(–ta < < ta) = 1 – a leads to the interval • The confidence interval for m (unknown s) is

  10. Chapter 8 – Confidence Interval for a Mean (m) with Unknown s Graphically…

  11. x = 510 s = 73.77 Chapter 8 – Confidence Interval for a Mean (m) with Unknown s Example: GMAT Scores • Here are GMAT scores from 20 applicants to an MBA program: • Construct a 90% confidence interval for the mean GMAT score of all MBA applicants. • Since s is unknown, use the Student’s t for the confidence interval with n = 20 – 1 = 19 d.f. • First find t0.90 from Appendix D.

  12. Chapter 8 – Confidence Interval for a Mean (m) with Unknown s Example: GMAT Scores For n = 20 – 1 = 19 d.f., t0.90 = 1.729 (from Appendix D).

  13. sn sn x - t x + t 510 + 28.52 < m < 510 – 28.52 73.77 20 73.77 20 510 – 1.729 510 + 1.729 < m < < m < Chapter 8 – Confidence Interval for a Mean (m) with Unknown s Example: GMAT Scores • The 90% confidence interval is: • We are 90% certain that the true mean GMAT score is within the interval 481.48 < m < 538.52.

  14. Chapter 8 – Confidence Interval for a Mean (m) with Unknown s Using Appendix D: • Beyond n = 50, Appendix D shows n in steps of 5 or 10. • If the table does not give the exact degrees of freedom, use the t-value for the next lower n. • This is a conservative procedure since it causes the interval to be slightly wider. Using MegaStat: • MegaStat gives you a choice of z or t and does all calculations for you.

  15. Clickers A sample of 20 final exams from BCOR1020 has an average of and a standard deviation of S = 8. Assuming the population of final exam scores is normally distributed, find the appropriate t distribution critical value to determine the 95% confidence interval for the mean score. (t-dist. Table on overhead) A) t = 1.725 B) t = 1.729 C) t = 2.086 D) t = 2.093

  16. Clickers A sample of 20 final exams from BCOR1020 has an average of and a standard deviation of S = 8. Assuming the population of final exam scores is normally distributed, find the 95% confidence interval for the mean score. (t-dist. Table on overhead) A) B) C) D)

  17. Chapter 8 – Confidence Interval for a Mean (m) with Unknown s Confidence Interval Width: • Confidence interval width reflects - the sample size, - the confidence level and - the standard deviation. • To obtain a narrower interval and more precision- increase the samplesize or - lower the confidence level (e.g., from 90% to 80% confidence)

  18. Chapter 8 – Sample Size Determination for C. I. for a Mean (m) • To estimate a population mean with a precision of +E (allowable error), you would need a sample of what size? Sample Size to Estimate m(assuming s is known): • This formula is derived by comparing the intervals m+E and m+ . • Solving the following equation for n: • Gives us the formula: Always round up!

  19. Method 1: Take a Preliminary Sample: Take a small preliminary sample and use the sample s in place of s in the sample size formula. Method 2: Assume Uniform Population: Estimate rough upper and lower limits a and b and set s = [(b-a)/12]½. Method 3: Assume Normal Population: Estimate rough upper and lower limits a and b and set s = (b-a)/4. This assumes normality with most of the data with m+ 2s so the range is 4s. Method 4: Poisson Arrivals: In the special case when m = l is a Poisson arrival rate, then s = m = l . Chapter 8 – Sample Size Determination for C. I. for a Mean (m) How to Estimate an unknown s?

  20. Chapter 8 – Sample Size Determination for C. I. for a Mean (m) Example: GMAT Scores • Recall from our last lecture that the population standard deviation for GMAT scores is s = 86.8. • How large should our sample be if we would like to have a 95% confidence interval for average GMAT scores that is no wider than + 20 points? • We know • s = 86.8, E = 20, z = 1.96 for 95% C.I. • So, • Since we always round up, we will draw a sample of size n = 73.

  21. Clickers Suppose we want to find a confidence interval for the mean score on BCOR1020 final exams. Assuming the population of final exam has a standard deviation of s = 8, how large should our sample be if we would like a 95% confidence interval no wider than + 2 points? A) n = 30 B) n = 62 C) n = 100 D) n = 246

More Related